Spectroscopic Data as a Discretized Manifold
PHyicalTDA operates on datasets of the general form $I(\mathbf{q},\omega)$ where $\hbar\mathbf{q}$ denotes momentum transfer and $\hbar\omega$ denotes energy transfer.
In neutron scattering experiments, this intensity represents a measured spectral density which encodes the allowed collective excitations of the system. From the perspective of PHysicalTDA, the dataset is treated as:
- A scalar field defined over a discretized parameter space
- Independent of the microscopic oigin of the signal (phonons, magnons, continua, etc.)
- A sampled representation of an underlying excitation manifold embedded in $(\mathbf{q},\omega)$ space
The Cubical Complex
Experimental datasets are naturally sampled on rectilinear grids due to detector geometry, binning, or interpolation. Accordingly, PHysicalTDA represents intensity data using a cubical complex:
- Each voxel corresponds to a sampled vaue of $I(\mathbf{q},\omega)$
- The grid defines adjacency relations used to construct the topological complex
- The scalar field values are used to define a filtration over the complex
The cubical complex is preferred over simplicial meshes (Vietoris-Rips, alpha, etc.) because it preserves the native structure of experimental data and avoids unnecessary triangulation artifacts.
Filtrations on Intensity Data
Persistent homology is computed by applying a filtration to the cubical complex covering the intensity data. In PHysicalTDA:
- Filtrations are either superlevel or sublevel
- Persistence tracks when a topological feature appears and vanishes
- Filtration parameters are chosen to be invariant under monotonic rescaling of intensity
This construction allows robust identification of topological features corresponding to excitation branches, gaps, continua, and intersections.
Excitation Manifolds
From a geometric perspective, dispersive excitations trace out lower-dimensional manifolds embedded in the higher-dimensional $(\mathbf{q},\omega)$ space. Persistent homology provides a way to characterize these structures without fitting dispersion relations or assuming symmetries. Topological features can describe connectivity of excitation branches, band merging/splitting, and the stability of features under noise/resolution effects.